How can we justify dropping the absolute time hypothesis? In some approaches to Special Relativity the theory is motivated talking about the Michelson-Morley experiment and how this relates to the postulate that the speed of light is the same in every inertial reference frame. Once someone has this postulate, it is quite simple to argue tha time cannot be absolute. This leads then to a new viewpoint about space and time and everything follows.
Another approach, about which I ask here is much more geometrical and has a nicer mathematical structure already prepared to generalize to General Relativity. The motivation for this geometrical viewpoint is just to consider Special Relativity as Galilean Relativity with absolute time hypothesis relaxed. In that case we derive the Lorentz transformations quite easily and everything follows.
Although this motivates the mathematical structure, I still cannot understand how to motivate this quite drastic step on dropping absolute time. I mean, absolute time is something quite natural, and it is so natural to think about it that Galileo, Newton and others before Einstein always did so.
The Michelson-Morley experiment motivates this step, but it's quite messy: we first postulate the constancy of the speed of light. Then with a thought experiment we show that simultaneous events in one reference frame are not in general simultaneous in another frame. Then we argue that time shouldn't be absolute at all.
Now is there another way to motivate dropping absolute time assumption? I mean, what is the real situations that shows us that we need to drop the absolute time assumption and how can we intuitively see that dropping it will solve the problems at hand?
 A: How can we justify dropping the absolute time hypothesis?
By simply looking at what clocks do. A clock doesn't actually "measure the flow of time". It isn't some cosmic gas meter. The passage of time is just a figure of speech. A clock clocks up some kind of regular cyclical local motion and displays a cumulative result that we call the time. Time is a dimension of measure, not a dimension that offers freedom of motion. I can hop forward a metre but you can't hop forward a second. Or backwards. Whilst you could take an out-and-back trip through space and suffer time dilation, all that really happens is that your local motion is reduced because of your macroscopic motion through space relative to me. The Lorentz factor is just Pythagoras's theorem. And that time dilation isn't time travel. I could have watched you every moment through my telescope, and you don't come back to the middle of last week.
Have a look at The Other Meaning of Special Relativity by Robert Close. It explains why the SR postulate works - the wave nature of matter means we calibrate our rods and clocks using the motion of waves, then we use them to measure the motion of waves. Then have a look at A World without Time: The Forgotten Legacy of Godel and Einstein by Palle Yourgrau. Once the penny drops that time is just a measure of local motion, you can justify dropping more than the absolute time hypothesis. Then it's like pulling a thread with Einstein's name on it, and out comes a string of pearls.        
A: It is not just absolute time that is being thrown out the window, it is also absolute rest and absolute motion being tossed as well. But to start, it must be understood that Time is a dimension. It is 1 dimension of the 4 dimensional structure known as Space-Time. If you are moving across the dimension of time, then time passes by. If however, you were able to be only moving across one of the three dimensions of space instead, then time would be at a standstill. 
But it is to be noted that if you examine the consequences of there being "absolute motion" taking place within an "absolute 4 dimensional environment" known as Space-Time, these consequences lead to the creation of the very same bizarre outcomes that are described under the title of Special Relativity.
This examination also leads to the creation of all of the SR equations, meaning the Lorentz-Fitzgerald Length Contraction equation, the Time Dilation equation, the Velocity Addition equation, and the Lorentz Transformation equations.
Meanwhile however, Special Relativity itself can not lead you to the "absolute motion" that is taking place within the "absolute 4 dimensional environment" known as Space-Time. As a consequence, both are ignored or are often assumed to simply not exist in the absolute sense.
A: If it is admissible the question and answer to 'How old is the universe ?' then we can not drop the absolute time notion.  
I can elaborate on other ways but one reason is enough to justify my answer.  
A: How about:


*

*Mathematically it's an interesting thing to do and a fairly obvious direction for generalizing the Galileo group, as I discuss in my other answer to you - it leads to the nontrivially signatured orthogonal groups; 

*Experimentally we have heaps of evidence for relaxing the absolute time postulate;

*Watch this video: "The Illusion of Time" by Brian Greene at about 10 minutes onwards.
Although the Greene video is obviously at quite a different level from what you are seeking, at about 10 minutes onwards he makes an important historical and cultural point which I had never thought about. Although I and probably Greene wouldn't claim to be real, rigorous historians, nonetheless I find the following highly plausible. Before the coming of high speed train travel, there simply was not the need to synchonize clocks precisely. Every town had its own local time and it really didn't make any difference to life if each town's timepieces varied a little. So the point is that up until this time, the concept of time was very coarse and thought of in terms of night and day, our birth, aging and dying with the heavy religious baggage that bears with it and so forth. There was no thought given to time other than very coarsely and the religious conceptions of its passing according to the will of the gods. In other words humans do NOT naturally have a sound intuition for the physical implications of time, other than very roughly. No one had ever taken the notion of relative time seriously, because nothing we did probed synchronization issues deeply. Religious dogma quickly filled this intuitional void, and so we have a deep cultural programming that time is absolute. But from a physics point of view, if you brush this cultural programming aside, you can sit back and think: there really is very little compelling evidence to assume anything for the purposes of physics about time until we get to the:


*

*Tight synchronization between timepieces that the industrial revolution required (amongst whcih trains sharing tracks and needing to avoid collisions) of the late 19th century

*Boosted metastable particle and atomic clock experiments of the 20th century that showed absolute time to be untenable.


Indeed Greene argues that Einstein's employment in the Swiss patent office honed his insight as he examined and thought carefully about the many time synchronization patents that crossed his desk. 
A: Galileo dropped the notion of absolute rest for reasons he described in a dialogue on two world systems; this also meant it wasn't possible to hold onto a notion of absolute motion (the notion was in his time seem as part of Aristotelian Philosophy; though Aristotle himself didn't hold it).
Newton then described space and time through his notions of absolute time and absolute space; that is we can name a specific time and specific position; he still held that there was no absolute motion.
Maxwell then showed that there was an absolute motion - the speed of light; this was a puzzle; and there was various attempts to fix this paradox through length contraction and time dilation by Lorentz and others.
The correct solution was eventually found by Poincare and Einstein by re-theorising what simultaneity meant through an operational definition; it turned out that simultaneity as a concept could only be made consistent with the absolute motion of light when there was no spatial seperation: ie it makes no physical sense to say that an event on Mars occurred at the same time as one on Earth (unless one fixes a frame).
Minkowski then suggested a geometric interpretation which melded space and time into the concept of a manifold with a metric; the metric described the motion of light; essentially he introduced the concept of absolute spacetime (not of space and time seperately).
A: I would turn the question around and ask why would you consider time to be absolute, since there is no evidence for it. And given that, why would you consider 'dropping' absolute time to be a drastic step?
The way we think about time before we get to learn about relativity is a product of our experience and evolution. The effects predicted by special relativity are so small at everyday speeds that we never notice them, so we get the idea that it is the same time everywhere (at least in our local time zone). And let's not forget that when we are kids we are brainwashed to believe that Euclidian geometry and Newton's laws describe the world. Given all that, it is hardly surprising that our intuitions are biased in a way that makes us scratch out heads when we are then confronted with special relativity. But in reality there is no real reason for assuming time is absolute.
In day to day life there's nothing that intuitively tells us we need to drop the assumption of absolute time, which is probably why we find it hard to let go of the idea. But if you accept that the speed of light is the same relative to everything, you have to adopt a different view of time.
To make the following illustration easier, let's assume the speed of light is 300,000,000 meters per second. Suppose we stand together and 300,000,000 metres to the left and right of us are light detectors with accurate clocks. If we flash a light, the detectors will tell us the light arrived at them a second later. If we repeat the experiment, but this time you walk to the right at 1m/s immediately after the flash, then the detectors will tell you that after a second, according to their clocks, the light was detected 299,999,999 metres to the right of you and 300,000,001 metres to the left of you. If the speed of light is constant, the only way you can explain that is to assume that the clocks on the detectors became out of synch relative to the time in your frame.
A: I have a thought model for absolute time using Interstellar: first Romilly and Cooper are on the Endurance spacecraft together, then Cooper goes to the planet Miller in the Gargantua system where the gravitational field make time flow much slower.  They are aware of this.  Given knowledge of the spacetime topography they could set adjusted timers so that Cooper's would ping every second and Romilly’s would ping every month, 'simultaneously'.  When they heard it ping they would know the other heard it at the same time.  Even without the spacemen, this time manifold stands.
Furthermore, even with multiple locations synchronised, time is only going to march forward.  The pings are always ordered.  When Cooper rejoins Romilly they are on the same ping.  No-one is going to be able to step into their own past, as Gödel suggests when he says the absence of a global time function

seems to imply an absurdity. For it enables one e.g., to travel into the near past of those places where he has himself
lived.

